Lazard's universal ring

In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in Lazard (1955) over which the universal commutative one-dimensional formal group law is defined.

There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let

${\displaystyle F(x,y)}$

be

${\displaystyle x+y+\sum _{i,j}c_{i,j}x^{i}y^{j}}$

for indeterminates ${\displaystyle c_{i,j}}$, and we define the universal ring R to be the commutative ring generated by the elements ${\displaystyle c_{i,j}}$, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring R has the following universal property:

For every commutative ring S, one-dimensional formal group laws over S correspond to ring homomorphisms from R to S.

The commutative ring R constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degree 1, 2, 3, ..., where ${\displaystyle c_{i,j}}$ has degree ${\displaystyle (i+j-1)}$. Daniel Quillen (1969) proved that the coefficient ring of complex cobordism is naturally isomorphic as a graded ring to Lazard's universal ring. Hence, topologists commonly regrade the Lazard ring so that ${\displaystyle c_{i,j}}$ has degree ${\displaystyle 2(i+j-1)}$, because the coefficient ring of complex cobordism is evenly graded.

References

• Adams, J. Frank (1974), Stable homotopy and generalised homology, University of Chicago Press, ISBN 978-0-226-00524-9
• Lazard, Michel (1955), "Sur les groupes de Lie formels à un paramètre", Bulletin de la Société Mathématique de France, 83: 251–274, doi:10.24033/bsmf.1462, MR 0073925
• Lazard, Michel (1975), Commutative formal groups, Lecture Notes in Mathematics, vol. 443, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070554, ISBN 978-3-540-07145-7, MR 0393050
• Quillen, Daniel (1969), "On the formal group laws of unoriented and complex cobordism theory", Bulletin of the American Mathematical Society, 75 (6): 1293–1298, doi:10.1090/S0002-9904-1969-12401-8, MR 0253350